Cycles having the same modularity and removable edges in 2-connected graphs Original Research Article

In this paper, we consider 2-connected multigraphs in which every cycle has length congruent to a modulo b (b⩾2). We prove that there exists such a multigraph which is homomorphic to a graph with minimum degree at least three only if a=0, and that there exists such a graph only if a=0 and b=2. We also study the distribution of paths whose internal vertices have degree exactly two, and show a relation between these paths and edges in a 2-connected graph whose deletion results in a 2-connected graph.